Since \(\left\lbrace u(X_t)-u(X_0)-\int _0^t{ \mathcal {L}}u(X_s)\,ds: t\ge 0\right\rbrace \)  is a
martingale (with respect to the natural filtration) for every \(u\in C_b^2({\mathbb {R}}^d)\) 
(cf. proof of [1]}), it follows that \(X\)  is the
process which corresponds to the kernel \(K(x,h)=k(h)\)  in our set-up.
